L(s) = 1 | + 2.82i·2-s + 3·3-s + 0.0423·4-s + 3.41i·5-s + 8.46i·6-s + 13.3i·7-s + 22.6i·8-s + 9·9-s − 9.62·10-s + 35.4i·11-s + 0.127·12-s − 37.6·14-s + 10.2i·15-s − 63.6·16-s − 69.6·17-s + 25.3i·18-s + ⋯ |
L(s) = 1 | + 0.997i·2-s + 0.577·3-s + 0.00529·4-s + 0.305i·5-s + 0.575i·6-s + 0.720i·7-s + 1.00i·8-s + 0.333·9-s − 0.304·10-s + 0.971i·11-s + 0.00305·12-s − 0.718·14-s + 0.176i·15-s − 0.994·16-s − 0.993·17-s + 0.332i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.292510525\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.292510525\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.82iT - 8T^{2} \) |
| 5 | \( 1 - 3.41iT - 125T^{2} \) |
| 7 | \( 1 - 13.3iT - 343T^{2} \) |
| 11 | \( 1 - 35.4iT - 1.33e3T^{2} \) |
| 17 | \( 1 + 69.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 12.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 126.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 179.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 255. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 207. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 117. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 553.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 62.9iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 147.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 274. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 603.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 741. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 572. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 26.7iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 207.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.22e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.79e3iT - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.00631561645618172920006200340, −9.780075797336613721673365856159, −8.925681349286439993054261486213, −8.175092930952444345785812931479, −7.14504521041284720927530524295, −6.65229355533410503162021979993, −5.46311077613530904885961522839, −4.49956362432734426708236560561, −2.87297032797859163600815532861, −1.96443017962581208481191897702,
0.62006450937489735566237366481, 1.77403267576419551389516123453, 3.07123743162878647497247200627, 3.80231722733161273055320085336, 5.02184138373851721664120505837, 6.58051382335413724943722003512, 7.29229226848941976211585999273, 8.591156847977326100995156901875, 9.185334434550752307798244944156, 10.32561068004749733907836001635